Differential cross sections and photon-beam asymmetries for the γ→p→K+Λ and γ→p→K+Σ0 reactions have been measured in the photon energy range from 1.5 to 2.4 GeV and in the angular range from Θc.m.=0° to 60° of the K+ scattering angle in the center-of-mass system at the SPring-8/LEPS facility. The photon-beam asymmetries for both the reactions have been found to be positive and to increase with the photon energy. The measured differential cross sections agree with the data measured by the CLAS Collaboration at cosΘc.m.<0.9 within the experimental uncertainties, but the discrepancy with the SAPHIR data for the K+Λ reaction is large at cosΘc.m.>0.9. In the K+Λ reaction, the resonance-like structure found in the CLAS and SAPHIR data at W=1.96 GeV is confirmed. The differential cross sections at forward angles suggest a strong K-exchange contribution in the t-channel for the K+Λ reaction, but not for the K+Σ0 reaction.